**Euler's Formula**

#e^{i theta}=cos theta + i sin theta#

Let us first review some useful power series.

#e^x=1/{0!}+x/{1!}+x^2/{2!}+cdots#

#cos x=1/{0!}-x^2/{2!}+x^4/{4!}-cdots#

#sin x=x/{1!}-x^3/{3!}+x^5/{5!}-cdots#

Now, we are ready to prove Euler's Formula.

**Proof**

By rewriting as a power series,

#e^{i theta}=1/{0!}+(i theta)/{1!}+(itheta)^2/{2!}+(i theta)^3/{3!}+(i theta)^4/{4!}+(i theta)^5/{5!}+cdots#

by distributing the powers,

#=1/{0!}+i theta/{1!}+i^2 theta^2/{2!}+i^3 theta^3/{3!}+i^4 theta^4/{4!}+i^5 theta^5/{5!}+cdots#

by #i^2=-1#

#=1/{0!}+i theta/{1!}-theta^2/{2!}-i theta^3/{3!}+theta^4/{4!}+i theta^5/{5!}-cdots#

by separating the real part and the imaginary part,

#=(1/{0!}-theta^2/{2!}+theta^4/{4!}-cdots)+i(theta/{1!}-theta^3/{3!}+theta^5/{5!}-cdots)#

by identifying the power series,

#=cos theta + i sin theta#

Hence, we have Euler's Formula

#e^{i theta}=cos theta+i sin theta#.

I hope that this was helpful.